Euclid,

ElementsV 22©

translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 22: If however-many magnitudes and others equal to them in plêthos, taken by two's, are also in the same ratio, they will be through an equal (ex aequali) in the same ratio.

(diagram 1) Let there be however-many magnitudes, A, B, G, and others equal to them in plêthos, D, E, Z, taken by two's in the same ratio, as A to B so D to E, and as B to G so E to Z. I say they will be through an equal (ex aequali) in the same ratio.

(diagram 2) For let there be taken of A, D equally-times multiples, H, Q, (diagram 3) and of B, E others, as it happens, equally-tiems multiples, K, L, (diagram 4 = gen. diag.) and furthermore of G, Z others, as it happens, equally-times multiples, M, N. (diagram 5) And since it is: as A to B so D to E, and there was taken of A, D, equally-times multiples, H, Q, and of B, E others, as it happens, equally-times multiples, K, L, (v 15) therefore it is: as H to K so Q to L. (v 4) (diagram 6) For the same reasons, in fact, as K to M so too L to N. (diagram 7) And so, since H, K, M are three magnitudes, and others equal to them in plêthos, Q, L, N, taken by two's and in the same ratio, therefore through an equal (ex aequali) if H exceeds M, Q also exceeds N, and if equal equal, and if smaller smaller. (v 20) (diagram 8) And H, Q are equally-times multiples of A,D, while M, N, are others , as it happens, equally-times mutiples of G, Z. (diagram 9) Therefore, as A to G so D to Z. Therefore, if however-many magnitudes and others equal to them in plêthos, taken by two's, are also in the same ratio, they will be through an equal (ex aequali) in the same ratio, just what it was required to show.